The main aim of this book is to present recent results concerning inequalities of the Jensen, Čebyšev and Grüss type for continuous functions of bounded selfadjoint operators on complex Hilbert spaces.
In the introductory chapter, the author portrays fundamental facts concerning bounded selfadjoint operators on complex Hilbert spaces. The generalized Schwarz’s inequality for positive selfadjoint operators as well as some results for the spectrum of this class of operators are presented. This text introduces the reader to the fundamental results for polynomials in a linear operator, continuous functions of selfadjoint operators as well as the step functions of selfadjoint operators. The spectral decomposition for this class of operators, which play a central role in the rest of the book and its consequences are introduced. At the end of the chapter, some classical operator inequalities are presented as well.
Recent new results that deal with different aspects of the famous Jensen operator inequality are explored through the second chapter. These include but are not limited to the operator version of the Dragomir-Ionescu inequality, the Slater type inequalities for operators and its inverses, Jensen’s inequality for twice differentiable functions whose second derivatives satisfy some upper and lower bound conditions and Jensen’s type inequalities for log-convex functions. Hermite-Hadamard’s type inequalities for convex functions and the corresponding results for operator convex functions are also presented.
The Čebyšev, (Chebyshev) inequality that compares the integral/discrete mean of the product with the product of the integral/discrete means is famous in the literature devoted to Mathematical Inequalities. The sister inequality due to Grüss which provides error bounds for the magnitude of the difference between the integral mean of the product and the product of the integral means has also attracted much interest since it has been discovered in 1935 with more than 200 papers published so far. The last part of the book is devoted to the operator versions of these famous results for continuous functions of selfadjoint operators on complex Hilbert spaces. Various particular cases of interest and related results are presented as well.
This book is intended for use by both researchers in various fields of Linear Operator Theory and Mathematical Inequalities, domains which have grown exponentially in the last decade, as well as by postgraduate students and scientists applying inequalities in their specific areas.
From the reviews:
“All the results presented in the book are proved along with the details of the original references where they were first obtained. ... Each Chapter is followed by a list of references used therein, and therefore they are relatively independent and can be read separately. ... I am pleased to recommend the book to readers interested in the latest developments in mathematical and functional analysis, a field of modern mathematics that has grown exponentially in the last few decades.” (George A. Anastassiou, SIAM Review, Vol. 55 (1), 2013)
“The aim of this monograph is to introduce several extensions of classical inequalities to inequalities of bounded linear operators on a complex Hilbert space. ... This monograph gives a good introduction into basic results in operator theory, and is suitable for undergraduate students. The author presents several operator inequalities, using similar techniques over and again. In this way, the readers may study ‘how to develop mathematics’ by reading the monograph.” (Takeaki Yamazaki, Zentralblatt MATH, Vol. 1242, 2012)
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